Question

Suppose C and D represent two different school populations where C > D and C and D must be greater than 0. Which of the following expressions is the largest? Explain why. Show all work necessary.

A.2(C + D)

B.(C + D)²

C.C² + D²

D.C² − D²

Answer 1

The expression (C + D)² is the largest because it includes the sum of the squares C² and D² plus an additional positive term 2CD, which makes it larger than all the other options. Option B is the correct answer.

Step-by-step explanation:

We are given two different school populations C and D where C > D and both are greater than 0. We need to determine which of the given expressions is the largest:

1. 2(C + D)
2. (C + D)²
3. C² + D²
4. C² − D²

Comparing option A and D first:

2(C + D) will yield a smaller value compared to C² + D² since squaring is a stronger operation than multiplying by 2 when dealing with positive numbers.
Hence, option A < option C.

Now, comparing option C and D, by rearranging option C (C² + D²) and adding and subtracting D²:

C² + D² = C² - D² + 2D²

Since D is a positive number, 2D² is positive, and therefore, option C is always larger than option D.

Finally, comparing option B and option C:

The square of a sum, (C + D)², is the same as C² + 2CD + D². Given that 2CD is a positive term because both C and D are positive, (C + D)² will be larger than C² + D².

Thus, the expression (C + D)² is the largest because it combines the two initial populations with an additional positive term that is based on the product of the two populations.

The correct option is B.